#4414503

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,4)(6,8)(9,10,14,15)(11,16,18,12)(13,17,22,23)(19,24,21,20)(27,33)(28,35)(30,36)(31,38);(10,15)(11,18)(17,23)(19,21)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(5,6,7,8)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,22)(23,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,2)(3,4)(9,12,14,16)(10,11,15,18)(13,20,22,24)(17,19,23,21)(25,27,29,33)(26,30,32,36)(28,34,35,39)(31,37,38,40);(9,13,14,22)(10,17,15,23)(11,19,18,21)(12,20,16,24)(25,28)(26,31)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4321699);